A simple and efficient numerical method for computing the dynamics of rotating Bose-Einstein condensates via a rotating Lagrangian coordinate

摘要

We propose a simple, efficient and accurate numerical method for simulatingthe dynamics of rotating Bose-Einstein condensates (BECs) in a rotational framewith/without a long-range dipole-dipole interaction. We begin with thethree-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentumrotation term and/or long-range dipole-dipole interaction, state thetwo-dimensional (2D) GPE obtained from the 3D GPE via dimension reduction underanisotropic external potential and review some dynamical laws related to the 2Dand 3D GPE. By introducing a rotating Lagrangian coordinate system, theoriginal GPEs are re-formulated to GPEs without the angular momentum rotationwhich is replaced by a time-dependent potential in the new coordinate system.We then cast the conserved quantities and dynamical laws in the new rotatingLagrangian coordinates. Based on the new formulation of the GPE for rotatingBECs in the rotating Lagrangian coordinates, a time-splitting spectral methodis presented for computing the dynamics of rotating BECs. The new numericalmethod is explicit, simple to implement, unconditionally stable and veryefficient in computation. It is spectral order accurate in space andsecond-order accurate in time, and conserves the mass in the discrete level.Extensive numerical results are reported to demonstrate the efficiency andaccuracy of the new numerical method. Finally, the numerical method is appliedto test the dynamical laws of rotating BECs such as the dynamics of condensatewidth, angular momentum expectation and center-of-mass, and to investigatenumerically the dynamics and interaction of quantized vortex lattices inrotating BECs without/with the long-range dipole-dipole interaction.